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[[144,12,16]] d ≤
n
144
k
12
d
16
kd²/n
21.333
w
9
X/Z
1

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Distance

X/Z asymmetry 1 · d_X ≤ 16, d_Z ≤ 16 · w_X = 9, w_Z = 9 (max(d_X,d_Z)/min(d_X,d_Z); each side carries its own earned tier: = certified exact, ≤ witness upper bound)
d_X 16 · witness weight 16 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 16 qubits)
[3, 5, 15, 26, 49, 58, 60, 64, 73, 91, 93, 98, 105, 106, 111, 122]
d_Z 16 · witness weight 16 (claimed upper_bound)
witness found by @mathysrennela · research/kit/surrogate.lightest_logical · found at 2000 trials · survived 2000 trials · 2026-10-12
witness operator (support, 16 qubits)
[27, 37, 46, 49, 54, 57, 63, 67, 82, 90, 98, 100, 108, 110, 119, 127]
certificate none yet · distance stands as a self-certified upper bound (d ≤)

Diagnostics

computed by the verifier from the parity checks, the layout, and the stored witnesses; shown as evidence, not used for ranking
girth H_X 4 · H_Z 4 (shortest cycle of each side’s Tanner graph; longer is friendlier to belief propagation)
check weights H_X 9 · H_Z 9
qubit degrees H_X 4–5 (mean 4.5) · H_Z 4–5 (mean 4.5)
trapping sets H_X (1,4)×72 (2,5)×72 (3,5)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_X
(1,4): 72 (1,5): 72 (2,5): 72 (2,6): 504 (2,7): 1296 (2,8): 576 (3,5): 72 (3,6): 432 (3,7): 2736 (3,8): 6912 (3,9): 19368 (3,10): 23328 (3,11): 6408 (3,12): 1800 (3,13): 360
trapping sets H_Z (1,4)×72 (2,5)×72 (3,5)×72 (smallest syndrome weight at each size, connected sets of up to 3 qubits)
full (size, syndrome weight): count census for H_Z
(1,4): 72 (1,5): 72 (2,5): 72 (2,6): 504 (2,7): 1296 (2,8): 576 (3,5): 72 (3,6): 432 (3,7): 2736 (3,8): 6912 (3,9): 19368 (3,10): 23328 (3,11): 6408 (3,12): 1800 (3,13): 360

Construction & provenance

authors Lee, Gideon
provenance literature baseline
construction Two-block group-algebra code C(a,b) over C12 x C6 = SmallGroup(72, 36) (order 72), elements numbered as in GAP Elements(SmallGroup(order,id)) with element 0 the identity. n = 2|G| = 144; H_X = [L(a)|R(b)], H_Z = [R(b)^T|L(a)^T]; a = {0, 1, 11, 28, 69}, b = {0, 15, 37, 50} as element indices. Check weight 9. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race.
model Mimo-V2.6-Flash (claimed, not verified)
reference arXiv:2610.13880
date 2026-10-12
notes Parameters and construction taken from the public database of arXiv:2610.13880v1. Distance is an upper bound certified by the witnesses attached here. Parameters and blocks taken from the public code database of arXiv:2610.13880v1 (Lee, 'Bicycle Race'), github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0; rebuilt here with the repo's own group data. Distance is an upper bound certified by the attached witnesses, not the paper's exact distance.
family generalized bicycle (a tag, not a ranking)
locality unrestricted (computed from the layout)
weight class weight > 8 (computed)

How this code was found

the research note submitted with this code · raw markdown · all notes

[[144,12,16]] — two-block group-algebra code over C12 x C6

Direction & hypothesis

Cell weight-9plus x unrestricted, CSS board. The argument is coverage, not discovery: this code is published in arXiv:2610.13880v1 (Lee, "Bicycle Race"), and nothing on the board occupied this (n, k, d, w) point. It advances the four-axis Pareto frontier by moving into a cell that was empty at these parameters.

What was searched

Not a search for new codes. The arXiv:2610.13880v1 database (github.com/gideonlsx/bicycle-race @ aed705a, CC BY 4.0, 12,448 records) was rebuilt here from its group and block data, then screened against every CSS entry on main under the site's Pareto rule (n <=, k >=, d >=, w <= with at least one strict) within the same weight class. Records whose parameters the paper already tabulates, and candidates claimed by other open filings, were removed first. Distances were re-searched locally:

  • X side: research/kit/surrogate.lightest_logical at 2,000 trials
  • Z side: research/kit/surrogate.lightest_logical at 2,000 trials

Evidence trail

Witness-backed upper bound, not an exact distance. The attached X and Z operators were found by the searches above and re-checked by the verifier's own criteria (in ker(opposite) and outside rowspace(own)). verify/validate_candidate.py reports passed: true and board_advancing: true, with d_only_gain: false.

The paper states an exact distance for this record, verified there with a second exact engine plus GAP rebuild and independent upper-bound attacks. That exactness is not claimed here; the board records what the witness certifies.

Dead ends

  • Direct sums of two board codes: rejected, CONTRIBUTING.md requires a single

    • connected Tanner graph and a connected stabilizer group.

  • Fitting a single-layer 2D layout to reach the local-2d cells: searched with

    • research/local2d/fold_layout.search_layout, best radii 10–13 against a ceiling of 4.0, so these codes stay unrestricted.

  • Hypergraph products of two parents from this pool: connectivity passes, but n

    • multiplies (two ~250-qubit parents give n = 80,010).

Tools

Model: Mimo-V2.6-Flash (self-reported, provenance.model), driving OpenCode. Rebuild: research/kit/group_algebra.build_2bga from the database's data/groups.jsonl Cayley table. Witness search: research/kit/surrogate (numpy backend, 2,000 trials/side screening) with the gf2_fast accelerator for longer runs. Gate: research/kit/coordination.py gate, which runs verify/validate_candidate.py unchanged. Search run by @mathysrennela.

Reproduction

from group_algebra import build_2bga
# a and b are element indices from github.com/gideonlsx/bicycle-race @ aed705a data/codes.jsonl
HX, HZ = build_2bga(mul, a, b)   # mul from data/groups.jsonl, n = 2|G| = 144

Parameters are also spelled out in provenance.construction of codes/144-12-16.json.

Parity checks

X-checks 72 (max weight 9) · Z-checks 72 (max weight 9)
H_X (72 checks, sparse supports)
[0, 9, 22, 29, 47, 72, 102, 104, 143] [0, 1, 34, 39, 55, 73, 112, 114, 134] [2, 16, 20, 32, 37, 74, 89, 119, 141] [3, 12, 23, 41, 48, 75, 77, 121, 130] [2, 4, 25, 43, 63, 76, 122, 124, 127] [1, 5, 26, 44, 50, 77, 83, 85, 138] [2, 6, 24, 42, 49, 78, 97, 123, 127] [3, 7, 20, 52, 56, 79, 81, 101, 129] [4, 6, 8, 53, 67, 80, 98, 130, 132] [5, 9, 14, 36, 58, 81, 91, 93, 142] [5, 10, 33, 38, 54, 82, 84, 106, 117] [0, 11, 40, 52, 56, 83, 107, 114, 135] [6, 12, 13, 35, 57, 76, 84, 98, 131] [8, 13, 28, 42, 62, 85, 87, 135, 140] [10, 14, 30, 44, 59, 86, 89, 92, 136] [2, 7, 15, 60, 64, 72, 87, 99, 111] [7, 11, 16, 45, 70, 84, 88, 137, 139] [8, 12, 17, 46, 61, 89, 101, 103, 108] [9, 10, 18, 43, 63, 88, 90, 92, 116] [1, 11, 19, 60, 64, 85, 91, 117, 139] [12, 20, 21, 27, 45, 80, 92, 108, 137] [13, 17, 21, 38, 66, 93, 95, 120, 139] [14, 18, 22, 40, 65, 74, 94, 97, 140] [6, 15, 23, 31, 68, 73, 95, 109, 122] [15, 16, 19, 24, 71, 92, 96, 118, 141] [17, 20, 25, 33, 54, 97, 111, 113, 119] [15, 19, 26, 51, 55, 98, 100, 124, 133] [3, 17, 27, 57, 66, 81, 99, 102, 125] [0, 18, 28, 53, 67, 74, 86, 96, 100] [4, 18, 29, 58, 65, 77, 101, 132, 142] [5, 19, 30, 31, 68, 88, 93, 102, 118] [24, 26, 31, 49, 60, 103, 106, 110, 142] [10, 21, 25, 32, 70, 75, 104, 118, 128] [21, 27, 33, 50, 61, 100, 105, 107, 112] [11, 22, 28, 34, 69, 76, 78, 106, 120] [23, 24, 30, 35, 62, 74, 107, 123, 126] [23, 26, 30, 36, 59, 105, 108, 110, 132] [7, 25, 27, 37, 70, 72, 109, 112, 133] [1, 28, 38, 46, 61, 78, 94, 107, 110] [8, 28, 29, 39, 69, 81, 103, 111, 143] [9, 30, 40, 41, 48, 96, 104, 112, 126] [31, 35, 36, 41, 57, 82, 113, 116, 143] [4, 18, 32, 42, 71, 79, 114, 126, 136] [32, 33, 37, 43, 58, 83, 110, 115, 117] [19, 34, 38, 44, 51, 80, 84, 116, 128] [3, 35, 40, 45, 66, 78, 117, 131, 135] [13, 34, 39, 46, 67, 95, 118, 121, 137] [3, 36, 40, 47, 65, 82, 103, 115, 119] [14, 35, 36, 48, 68, 87, 97, 120, 122] [4, 15, 37, 49, 71, 73, 83, 105, 121] [17, 38, 39, 50, 51, 72, 113, 122, 134] [1, 46, 51, 64, 69, 119, 123, 125, 129] [27, 41, 45, 47, 52, 86, 90, 124, 134] [29, 42, 43, 49, 53, 82, 88, 91, 125] [21, 44, 46, 50, 54, 75, 126, 129, 141] [7, 11, 47, 55, 69, 90, 113, 125, 127] [22, 45, 47, 48, 56, 76, 95, 128, 130] [8, 23, 49, 57, 62, 77, 91, 115, 129] [10, 25, 50, 58, 59, 73, 124, 130, 138] [5, 51, 54, 59, 68, 99, 127, 131, 133] [16, 37, 52, 55, 60, 94, 100, 132, 138] [13, 39, 53, 57, 61, 90, 96, 102, 133] [6, 31, 53, 62, 71, 104, 116, 134, 136] [14, 32, 54, 58, 63, 79, 99, 123, 135] [16, 34, 55, 56, 64, 75, 80, 101, 136] [9, 48, 59, 63, 65, 98, 105, 109, 137] [12, 41, 61, 62, 66, 86, 114, 138, 140] [22, 29, 42, 63, 67, 87, 109, 131, 139] [24, 26, 44, 64, 68, 79, 89, 111, 140] [0, 56, 65, 67, 69, 108, 115, 121, 141] [20, 33, 52, 66, 70, 85, 94, 120, 142] [2, 43, 60, 70, 71, 93, 106, 128, 143]
H_Z (72 checks, sparse supports)
[0, 15, 37, 50, 72, 73, 83, 100, 141] [1, 23, 49, 58, 73, 77, 91, 110, 123] [2, 22, 28, 35, 74, 76, 78, 87, 143] [3, 32, 54, 64, 75, 79, 99, 117, 119] [4, 12, 34, 56, 76, 80, 101, 114, 121] [3, 5, 29, 57, 77, 81, 82, 102, 131] [6, 34, 38, 45, 78, 80, 84, 95, 134] [7, 42, 63, 68, 79, 87, 88, 109, 127] [8, 20, 44, 64, 80, 85, 89, 111, 129] [7, 9, 27, 39, 72, 81, 90, 112, 137] [10, 41, 47, 53, 82, 86, 90, 104, 130] [5, 11, 43, 49, 83, 88, 91, 106, 127] [10, 12, 16, 44, 75, 84, 89, 92, 138] [5, 13, 19, 70, 84, 85, 93, 118, 133] [14, 28, 52, 66, 81, 86, 94, 120, 135] [13, 15, 48, 67, 87, 95, 96, 98, 121] [16, 18, 30, 53, 74, 88, 96, 132, 136] [2, 14, 17, 68, 89, 93, 97, 99, 122] [18, 52, 55, 61, 90, 94, 100, 101, 114] [9, 19, 53, 57, 91, 96, 98, 102, 116] [14, 18, 20, 24, 74, 79, 92, 97, 142] [9, 21, 30, 71, 92, 93, 104, 105, 126] [22, 38, 60, 70, 72, 94, 106, 128, 139] [21, 23, 46, 56, 75, 95, 107, 108, 129] [24, 28, 40, 61, 78, 96, 103, 107, 140] [6, 22, 25, 48, 76, 97, 104, 109, 130] [8, 12, 26, 65, 77, 98, 103, 108, 140] [15, 27, 59, 63, 92, 99, 105, 109, 124] [26, 28, 33, 60, 85, 100, 106, 110, 111] [7, 17, 29, 64, 72, 101, 111, 125, 139] [0, 27, 30, 61, 86, 102, 107, 108, 112] [17, 31, 39, 47, 95, 102, 103, 113, 134] [0, 32, 40, 62, 74, 104, 114, 115, 135] [33, 36, 49, 65, 82, 97, 105, 115, 142] [10, 31, 34, 71, 73, 106, 116, 118, 136] [11, 33, 35, 38, 84, 107, 113, 117, 120] [17, 20, 36, 69, 81, 108, 113, 119, 120] [23, 37, 65, 67, 74, 109, 115, 121, 132] [31, 36, 38, 43, 82, 93, 110, 116, 122] [15, 25, 39, 68, 73, 111, 118, 122, 133] [1, 33, 37, 40, 83, 94, 112, 117, 119] [25, 41, 50, 55, 75, 112, 113, 124, 138] [1, 11, 42, 66, 78, 85, 114, 125, 139] [43, 47, 57, 69, 76, 90, 115, 125, 143] [18, 41, 44, 62, 77, 86, 116, 126, 140] [10, 19, 43, 45, 88, 92, 117, 124, 128] [24, 30, 32, 46, 89, 110, 118, 123, 126] [2, 25, 47, 51, 72, 119, 124, 127, 128] [21, 34, 48, 70, 75, 112, 120, 128, 137] [3, 46, 49, 69, 78, 103, 121, 125, 129] [4, 23, 48, 50, 77, 105, 122, 126, 130] [6, 35, 51, 63, 98, 116, 122, 123, 131] [4, 26, 52, 58, 79, 83, 124, 132, 142] [27, 51, 53, 55, 80, 100, 125, 133, 134] [35, 40, 42, 54, 82, 97, 126, 131, 135] [4, 6, 55, 59, 73, 98, 127, 132, 136] [32, 44, 56, 71, 79, 83, 128, 136, 141] [7, 51, 54, 57, 84, 99, 113, 129, 133] [3, 8, 56, 58, 81, 101, 115, 130, 135] [12, 45, 59, 67, 86, 108, 130, 131, 137] [8, 29, 36, 60, 87, 91, 103, 132, 143] [26, 37, 59, 61, 89, 105, 110, 133, 138] [1, 50, 52, 62, 85, 107, 129, 134, 138] [11, 13, 45, 63, 76, 90, 135, 137, 139] [14, 42, 62, 64, 87, 91, 123, 136, 140] [16, 20, 46, 65, 94, 101, 119, 137, 141] [5, 58, 60, 66, 93, 99, 117, 138, 142] [16, 19, 21, 67, 80, 100, 118, 139, 141] [13, 22, 66, 68, 95, 102, 120, 131, 140] [2, 24, 54, 69, 106, 111, 123, 127, 141] [9, 29, 31, 70, 88, 104, 109, 142, 143] [0, 39, 41, 71, 96, 114, 121, 134, 143]
Code ID 144-12-16 · download JSON · raw on GitHub